Integrand size = 26, antiderivative size = 319 \[ \int \frac {1}{x \sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\frac {4 (2 b c+a d) \sqrt {c+d x}}{3 a^2 c^2 \sqrt {a x+b x^2}}-\frac {2 \sqrt {c+d x}}{3 a c x \sqrt {a x+b x^2}}+\frac {2 \sqrt {b} \left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) \sqrt {x} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{5/2} c^2 (b c-a d) \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}}-\frac {2 \sqrt {b} d (4 b c-a d) \sqrt {x} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 a^{3/2} c^2 (b c-a d) \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}} \] Output:
4/3*(a*d+2*b*c)*(d*x+c)^(1/2)/a^2/c^2/(b*x^2+a*x)^(1/2)-2/3*(d*x+c)^(1/2)/ a/c/x/(b*x^2+a*x)^(1/2)+2/3*b^(1/2)*(-2*a^2*d^2-3*a*b*c*d+8*b^2*c^2)*x^(1/ 2)*(d*x+c)^(1/2)*EllipticE(b^(1/2)*x^(1/2)/a^(1/2)/(1+b*x/a)^(1/2),(1-a*d/ b/c)^(1/2))/a^(5/2)/c^2/(-a*d+b*c)/(a*(d*x+c)/c/(b*x+a))^(1/2)/(b*x^2+a*x) ^(1/2)-2/3*b^(1/2)*d*(-a*d+4*b*c)*x^(1/2)*(d*x+c)^(1/2)*InverseJacobiAM(ar ctan(b^(1/2)*x^(1/2)/a^(1/2)),(1-a*d/b/c)^(1/2))/a^(3/2)/c^2/(-a*d+b*c)/(a *(d*x+c)/c/(b*x+a))^(1/2)/(b*x^2+a*x)^(1/2)
Result contains complex when optimal does not.
Time = 13.84 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\frac {-2 a c (c+d x) \left (a^2 d-4 b^2 c x+a b (-c+d x)\right )-2 i \sqrt {\frac {a}{b}} b d \left (-8 b^2 c^2+3 a b c d+2 a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{5/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )+4 i \sqrt {\frac {a}{b}} b d \left (-2 b^2 c^2+a b c d+a^2 d^2\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{5/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )}{3 a^3 c^2 (-b c+a d) x \sqrt {x (a+b x)} \sqrt {c+d x}} \] Input:
Integrate[1/(x*Sqrt[c + d*x]*(a*x + b*x^2)^(3/2)),x]
Output:
(-2*a*c*(c + d*x)*(a^2*d - 4*b^2*c*x + a*b*(-c + d*x)) - (2*I)*Sqrt[a/b]*b *d*(-8*b^2*c^2 + 3*a*b*c*d + 2*a^2*d^2)*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x) ]*x^(5/2)*EllipticE[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)] + (4*I)*Sqr t[a/b]*b*d*(-2*b^2*c^2 + a*b*c*d + a^2*d^2)*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/( d*x)]*x^(5/2)*EllipticF[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)])/(3*a^3 *c^2*(-(b*c) + a*d)*x*Sqrt[x*(a + b*x)]*Sqrt[c + d*x])
Time = 0.90 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.30, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {1261, 115, 27, 169, 27, 169, 27, 176, 122, 120, 127, 126}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x \left (a x+b x^2\right )^{3/2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 1261 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \int \frac {1}{x^{5/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {2 \int \frac {4 b c+2 a d+3 b d x}{2 x^{3/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a c}-\frac {2 \sqrt {c+d x}}{3 a c x^{3/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {\int \frac {2 (2 b c+a d)+3 b d x}{x^{3/2} (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 a c}-\frac {2 \sqrt {c+d x}}{3 a c x^{3/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {2 \int \frac {b (c (8 b c+a d)+2 d (2 b c+a d) x)}{2 \sqrt {x} (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c \sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x}}{3 a c x^{3/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {b \int \frac {c (8 b c+a d)+2 d (2 b c+a d) x}{\sqrt {x} (a+b x)^{3/2} \sqrt {c+d x}}dx}{a c}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c \sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x}}{3 a c x^{3/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {b \left (\frac {2 \int -\frac {d \left (a c (4 b c-a d)+\left (8 b^2 c^2-3 a b d c-2 a^2 d^2\right ) x\right )}{2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a (b c-a d)}+\frac {2 \sqrt {x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}\right )}{a c}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c \sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x}}{3 a c x^{3/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {b \left (\frac {2 \sqrt {x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}-\frac {d \int \frac {a c (4 b c-a d)+\left (8 b^2 c^2-3 a b d c-2 a^2 d^2\right ) x}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a (b c-a d)}\right )}{a c}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c \sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x}}{3 a c x^{3/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {b \left (\frac {2 \sqrt {x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {\left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {x} \sqrt {a+b x}}dx}{d}-\frac {c (b c-a d) (a d+8 b c) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a (b c-a d)}\right )}{a c}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c \sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x}}{3 a c x^{3/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {b \left (\frac {2 \sqrt {x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {\sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (b c-a d) (a d+8 b c) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a (b c-a d)}\right )}{a c}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c \sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x}}{3 a c x^{3/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {b \left (\frac {2 \sqrt {x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (b c-a d) (a d+8 b c) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a (b c-a d)}\right )}{a c}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c \sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x}}{3 a c x^{3/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {b \left (\frac {2 \sqrt {x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) (a d+8 b c) \int \frac {1}{\sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}\right )}{a (b c-a d)}\right )}{a c}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c \sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x}}{3 a c x^{3/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {x^{3/2} (a+b x)^{3/2} \left (-\frac {-\frac {b \left (\frac {2 \sqrt {x} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right )}{a \sqrt {a+b x} (b c-a d)}-\frac {d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {2 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) (a d+8 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {c+d x}}\right )}{a (b c-a d)}\right )}{a c}-\frac {4 \sqrt {c+d x} (a d+2 b c)}{a c \sqrt {x} \sqrt {a+b x}}}{3 a c}-\frac {2 \sqrt {c+d x}}{3 a c x^{3/2} \sqrt {a+b x}}\right )}{\left (a x+b x^2\right )^{3/2}}\) |
Input:
Int[1/(x*Sqrt[c + d*x]*(a*x + b*x^2)^(3/2)),x]
Output:
(x^(3/2)*(a + b*x)^(3/2)*((-2*Sqrt[c + d*x])/(3*a*c*x^(3/2)*Sqrt[a + b*x]) - ((-4*(2*b*c + a*d)*Sqrt[c + d*x])/(a*c*Sqrt[x]*Sqrt[a + b*x]) - (b*((2* (8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*Sqrt[x]*Sqrt[c + d*x])/(a*(b*c - a*d)* Sqrt[a + b*x]) - (d*((2*Sqrt[-a]*(8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*Sqrt[ 1 + (b*x)/a]*Sqrt[c + d*x]*EllipticE[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[-a]], ( a*d)/(b*c)])/(Sqrt[b]*d*Sqrt[a + b*x]*Sqrt[1 + (d*x)/c]) - (2*Sqrt[-a]*c*( b*c - a*d)*(8*b*c + a*d)*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c]*EllipticF[Arc Sin[(Sqrt[b]*Sqrt[x])/Sqrt[-a]], (a*d)/(b*c)])/(Sqrt[b]*d*Sqrt[a + b*x]*Sq rt[c + d*x])))/(a*(b*c - a*d))))/(a*c))/(3*a*c)))/(a*x + b*x^2)^(3/2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt Q[e, 0] && !LtQ[-b/d, 0]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) ) Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b , c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x ])) Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) ^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, n}, x] && !IGtQ[n, 0]
Time = 3.34 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.68
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (b x +a \right ) \left (d x +c \right )}\, \left (-\frac {2 \left (b d \,x^{2}+c b x \right ) b^{2}}{\left (a d -b c \right ) a^{3} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d \,x^{2}+c b x \right )}}-\frac {2 \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}{3 a^{2} c \,x^{2}}+\frac {2 \left (b d \,x^{2}+a d x +c b x +a c \right ) \left (2 a d +5 b c \right )}{3 a^{3} c^{2} \sqrt {x \left (b d \,x^{2}+a d x +c b x +a c \right )}}+\frac {2 \left (\frac {b^{2}}{a^{3}}+\frac {b^{3} c}{\left (a d -b c \right ) a^{3}}-\frac {d b}{3 a^{2} c}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}+\frac {2 \left (\frac {b^{3} d}{a^{3} \left (a d -b c \right )}-\frac {b d \left (2 a d +5 b c \right )}{3 c^{2} a^{3}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}\right )}{\sqrt {x \left (b x +a \right )}\, \sqrt {d x +c}}\) | \(537\) |
| default | \(\frac {2 \left (2 x \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} c \,d^{3}+2 x \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} b \,c^{2} d^{2}-4 x \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a \,b^{2} c^{3} d -2 x \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{3} c \,d^{3}-x \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} b \,c^{2} d^{2}+11 x \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a \,b^{2} c^{3} d -8 x \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{3} c^{4}+2 a^{2} d^{4} b \,x^{3}+3 a \,b^{2} c \,d^{3} x^{3}-8 x^{3} b^{3} c^{2} d^{2}+2 a^{3} d^{4} x^{2}+4 a^{2} b c \,d^{3} x^{2}-a \,b^{2} c^{2} d^{2} x^{2}-8 b^{3} c^{3} d \,x^{2}+x \,a^{3} c \,d^{3}+3 a^{2} b \,c^{2} d^{2} x -4 a \,b^{2} c^{3} d x -a^{3} c^{2} d^{2}+a^{2} b \,c^{3} d \right ) \sqrt {x \left (b x +a \right )}}{3 x^{2} \left (b x +a \right ) \left (a d -b c \right ) d \,a^{3} c^{2} \sqrt {d x +c}}\) | \(738\) |
Input:
int(1/x/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
(x*(b*x+a)*(d*x+c))^(1/2)/(x*(b*x+a))^(1/2)/(d*x+c)^(1/2)*(-2*(b*d*x^2+b*c *x)/(a*d-b*c)*b^2/a^3/((x+a/b)*(b*d*x^2+b*c*x))^(1/2)-2/3/a^2/c/x^2*(b*d*x ^3+a*d*x^2+b*c*x^2+a*c*x)^(1/2)+2/3*(b*d*x^2+a*d*x+b*c*x+a*c)/a^3/c^2*(2*a *d+5*b*c)/(x*(b*d*x^2+a*d*x+b*c*x+a*c))^(1/2)+2*(b^2/a^3+b^3*c/(a*d-b*c)/a ^3-1/3/a^2*d*b/c)*c/d*((x+c/d)/c*d)^(1/2)*((x+a/b)/(-c/d+a/b))^(1/2)*(-1/c *x*d)^(1/2)/(b*d*x^3+a*d*x^2+b*c*x^2+a*c*x)^(1/2)*EllipticF(((x+c/d)/c*d)^ (1/2),(-c/d/(-c/d+a/b))^(1/2))+2*(b^3/a^3*d/(a*d-b*c)-1/3*b*d*(2*a*d+5*b*c )/c^2/a^3)*c/d*((x+c/d)/c*d)^(1/2)*((x+a/b)/(-c/d+a/b))^(1/2)*(-1/c*x*d)^( 1/2)/(b*d*x^3+a*d*x^2+b*c*x^2+a*c*x)^(1/2)*((-c/d+a/b)*EllipticE(((x+c/d)/ c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))-a/b*EllipticF(((x+c/d)/c*d)^(1/2),(-c/ d/(-c/d+a/b))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 616 vs. \(2 (280) = 560\).
Time = 0.10 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.93 \[ \int \frac {1}{x \sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left ({\left (8 \, b^{4} c^{3} - 7 \, a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{3} + {\left (8 \, a b^{3} c^{3} - 7 \, a^{2} b^{2} c^{2} d - 2 \, a^{3} b c d^{2} - 2 \, a^{4} d^{3}\right )} x^{2}\right )} \sqrt {b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 3 \, {\left ({\left (8 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} x^{3} + {\left (8 \, a b^{3} c^{2} d - 3 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) - 3 \, {\left (a^{2} b^{2} c^{2} d - a^{3} b c d^{2} - {\left (8 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} x^{2} - 2 \, {\left (2 \, a b^{3} c^{2} d - a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a x} \sqrt {d x + c}\right )}}{9 \, {\left ({\left (a^{3} b^{3} c^{3} d - a^{4} b^{2} c^{2} d^{2}\right )} x^{3} + {\left (a^{4} b^{2} c^{3} d - a^{5} b c^{2} d^{2}\right )} x^{2}\right )}} \] Input:
integrate(1/x/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x, algorithm="fricas")
Output:
2/9*(((8*b^4*c^3 - 7*a*b^3*c^2*d - 2*a^2*b^2*c*d^2 - 2*a^3*b*d^3)*x^3 + (8 *a*b^3*c^3 - 7*a^2*b^2*c^2*d - 2*a^3*b*c*d^2 - 2*a^4*d^3)*x^2)*sqrt(b*d)*w eierstrassPInverse(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b ^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), 1/3*(3*b*d* x + b*c + a*d)/(b*d)) + 3*((8*b^4*c^2*d - 3*a*b^3*c*d^2 - 2*a^2*b^2*d^3)*x ^3 + (8*a*b^3*c^2*d - 3*a^2*b^2*c*d^2 - 2*a^3*b*d^3)*x^2)*sqrt(b*d)*weiers trassZeta(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), weierstrassPInverse( 4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c^3 - 3*a*b^2*c^ 2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), 1/3*(3*b*d*x + b*c + a*d)/(b*d ))) - 3*(a^2*b^2*c^2*d - a^3*b*c*d^2 - (8*b^4*c^2*d - 3*a*b^3*c*d^2 - 2*a^ 2*b^2*d^3)*x^2 - 2*(2*a*b^3*c^2*d - a^2*b^2*c*d^2 - a^3*b*d^3)*x)*sqrt(b*x ^2 + a*x)*sqrt(d*x + c))/((a^3*b^3*c^3*d - a^4*b^2*c^2*d^2)*x^3 + (a^4*b^2 *c^3*d - a^5*b*c^2*d^2)*x^2)
\[ \int \frac {1}{x \sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {1}{x \left (x \left (a + b x\right )\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \] Input:
integrate(1/x/(d*x+c)**(1/2)/(b*x**2+a*x)**(3/2),x)
Output:
Integral(1/(x*(x*(a + b*x))**(3/2)*sqrt(c + d*x)), x)
\[ \int \frac {1}{x \sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} \sqrt {d x + c} x} \,d x } \] Input:
integrate(1/x/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a*x)^(3/2)*sqrt(d*x + c)*x), x)
\[ \int \frac {1}{x \sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} \sqrt {d x + c} x} \,d x } \] Input:
integrate(1/x/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a*x)^(3/2)*sqrt(d*x + c)*x), x)
Timed out. \[ \int \frac {1}{x \sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {1}{x\,{\left (b\,x^2+a\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \] Input:
int(1/(x*(a*x + b*x^2)^(3/2)*(c + d*x)^(1/2)),x)
Output:
int(1/(x*(a*x + b*x^2)^(3/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {1}{x \sqrt {c+d x} \left (a x+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(1/x/(d*x+c)^(1/2)/(b*x^2+a*x)^(3/2),x)
Output:
( - 2*sqrt(c + d*x)*sqrt(a + b*x)*a*c + 4*sqrt(c + d*x)*sqrt(a + b*x)*a*d* x + 8*sqrt(c + d*x)*sqrt(a + b*x)*b*c*x + 2*sqrt(x)*int((sqrt(c + d*x)*sqr t(a + b*x)*x)/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*s qrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x)*a**2*b*d **2*x + 4*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x)*x)/(sqrt(x)*a**2*c + sq rt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c *x**2 + sqrt(x)*b**2*d*x**3),x)*a*b**2*c*d*x + 2*sqrt(x)*int((sqrt(c + d*x )*sqrt(a + b*x)*x)/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x)*a*b **2*d**2*x**2 + 4*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x)*x)/(sqrt(x)*a** 2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x )*b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x)*b**3*c*d*x**2 + sqrt(x)*int((sqrt( c + d*x)*sqrt(a + b*x))/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b *c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x )*a**2*b*c*d*x + 8*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a**2 *c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x) *b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x)*a*b**2*c**2*x + sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b* c*x + 2*sqrt(x)*a*b*d*x**2 + sqrt(x)*b**2*c*x**2 + sqrt(x)*b**2*d*x**3),x) *a*b**2*c*d*x**2 + 8*sqrt(x)*int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)...